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Hydraulics, open channels and fluid power
Euler head transferred by a pump impeller
Ideal head given to the fluid by an impeller, from the blade speeds U and tangential components Vθ of the absolute velocity at outlet (2) and inlet (1).
- Ref
- HYD-39
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- HE [m]
Applying angular momentum to the flow through an impeller gives the ideal energy transfer per unit mass, U₂Vθ₂ − U₁Vθ₁, with U the blade speed and Vθ the tangential component of the absolute velocity. Dividing by g gives the Euler head HE. With no whirl at the inlet it reduces to U₂Vθ₂/g. The real head is lower, because of friction, slip and other losses in the machine.
Ideal head given to the fluid by an impeller, from the blade speeds U and tangential components Vθ of the absolute velocity at outlet (2) and inlet (1). Actual head is lower, because of losses.
Calculator
- Outlet term, U₂Vθ₂/g24.46 m
- Inlet term, U₁Vθ₁/g0.8155 m
- Specific work, U₂Vθ₂ − U₁Vθ₁232 J/kg
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The impeller gives each kilogram 232 J, an ideal head of 23.65 m.
- Removing the inlet whirl (Vθ₁ = 0) would raise the head to 24.46 m, which is why pumps are designed for radial entry.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Vθ2 Outlet tangential velocity component | −50 to 200 m/s | −102.8 | 406.9 | 509.7 | +5 m/s | +43.1 % |
| U2 Blade speed at the outlet | 1 to 200 m/s | 0.4077 | 243.8 | 243.4 | +10% | +10.3 % |
| g Gravitational acceleration | 1 to 30 m/s² | 7.733 | 232 | 224.3 | +10% | −9.09 % |
| Vθ1 Inlet tangential velocity component | −50 to 200 m/s | −138.6 | 65.24 | 203.9 | +5 m/s | −17.2 % |
| U1 Blade speed at the inlet | 0 to 200 m/s | 4.077 | 24.46 | 20.39 | +10% | −0.345 % |
Sweep of Blade speed at the outlet
| Blade speed at the outlet (m/s) | Euler head (m) |
|---|---|
| 1 | 0.4077 |
| 25.88 | 30.84 |
| 50.75 | 61.26 |
| 75.63 | 91.69 |
| 100.5 | 122.1 |
| 125.4 | 152.5 |
| 150.3 | 183 |
| 175.1 | 213.4 |
| 200 | 243.8 |
Sweep one variable
Variables and how each one contributes
Euler head
What the equation gives you. Everything else feeds it.
Blade speed at the outlet
The blade speed at the outlet, in m/s. Raises the head, since it multiplies Vθ₂. Raising it by 10 % (to 22) moves the result by +10.3 %, from 23.65 to 26.1 m.
Outlet tangential velocity component
The tangential component of the absolute velocity at the outlet, in m/s. Raises the head in direct proportion to U₂. Raising it by 10 % (to 13.2) moves the result by +10.3 %, from 23.65 to 26.1 m.
Blade speed at the inlet
The blade speed at the inlet, in m/s. Lowers the head, through the inlet term. Raising it by 10 % (to 8.8) moves the result by −0.345 %, from 23.65 to 23.57 m.
Inlet tangential velocity component
The tangential component of the absolute velocity at the inlet, in m/s. Lowers the head, through the inlet term. Raising it by 10 % (to 1.1) moves the result by −0.345 %, from 23.65 to 23.57 m.
Gravitational acceleration
The gravitational acceleration, in m/s². Lowers the head in inverse proportion. Raising it by 10 % (to 10.79) moves the result by −9.09 %, from 23.65 to 21.5 m.
Worked example with the values above
- Outlet term: U₂Vθ₂ = 20 × 12 = 240 m²/s².
- Inlet term: U₁Vθ₁ = 8 × 1 = 8 m²/s².
- HE = (240 − 8) / 9.81 = 23.649 m.
Assumptions and limits
- Steady incompressible flow with the pump curve and system curve both of quadratic form.
- Ideal blade-guided flow for the velocity-triangle results, so the real head is lower.
Common mistakes
- Using gauge pressure where absolute pressure is needed for NPSH.
- Forgetting that pump curves in different units, such as litres per second, need converting to m³/s.
- Adding the flows of parallel pumps at different heads.
Statistics and uncertainty
Pump curves are known only to within a few percent and wear them down in service, and available suction head changes with the level, temperature and flow, so use a safety margin on the requirement.
Where it is used
- Choosing pumps and finding the duty point on a pipe system.
- Checking cavitation margin and estimating head from impeller geometry.
Maths you need
- Pump curve: head against flow for a pump at a fixed speed
- System curve: head needed by the pipe system against flow
- Velocity triangle: the vector diagram of blade and fluid velocities